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- W2789866470 abstract "The aim of this article is to survey ring-theoretic properties of Kasch, the regularity and the injectivity of the ring of real-continuous functions on a topoframe $L_{ tau}$, i.e., $mathcal{R}L_tau$. In order to study these properties, the concept of $P$-spaces and extremally disconnected spaces are extend to topoframes. For a $P$- topoframe $L_{ tau}$, the ring $mathcal{R}L_tau$ is $aleph_0$-Kasch ring. $P$- topoframes are characterized in terms of ring-theoretic properties of the regularity and injectivity of the ring of real-continuous functions on a topoframe. It follows from these characterizations that the ring $mathcal{R}L_tau$ is regular if and only if it is $aleph_0$-selfinjective. For a completely regular topoframe $L_tau$, we show that $mathcal{R}L_tau$ is a Bear ring if and only if it is a $CS$-ring if and only if $L_tau$ is extremally disconnected and also prove that it is selfinjective ring if and only if $L_{ tau}$ is an extremally disconnected $P$-topoframe." @default.
- W2789866470 created "2018-03-29" @default.
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- W2789866470 date "2018-03-15" @default.
- W2789866470 modified "2023-09-27" @default.
- W2789866470 title "Some ring-theoretic properties of the ring of $mathcal{R}L_tau$" @default.
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